Probing Excited $q\bar{q}$ Mesons via QCD Sum Rules

TL;DR

The paper addresses the challenge of identifying excited $q\bar{q}$ mesons by employing QCD sum rules with derivative-based interpolating currents. By using Gaussian sum rules at NLO and including up to dimension-8 condensates, the authors extract masses for $J^P=2^{\pm}$ nonets that largely agree with experimental states, and find compatible results for $J=0,1$ channels. A two-resonance Gaussian sum rule analysis is essential for resolving the $2^{++}$ sector, yielding two states consistent with known nonets and revealing which current couples preferentially to the heavier resonance. Overall, covariant-derivative operators prove to be a robust tool for studying excited hadrons, with implications for exploring exotic states beyond the conventional $q\bar{q}$ picture.

Abstract

We present a systematic study of the masses of light excited $q\bar{q}$ mesons using QCD sum rules at next-to-leading order (NLO). To probe excited states, we construct several interpolating currents involving covariant derivatives. The calculation is carried out up to dimension-8 condensates, including NLO perturbative and $m\langle\bar{q}q\rangle$ corrections. Employing Gaussian sum rules, we obtain several $J^P=2^\pm$ nonets with masses agreeing well with experiments. Several $J=0,1$ states compatible with experiments are also obtained using both Gaussian and Laplace sum rules. In particular, the $J^P=2^+$ current couples to two distinct $J^P=2^+$ resonances. This work demonstrates the efficacy of operators with covariant derivatives for studying excited hadrons.

Figures (10)

• Figure 1: The diagrams involved in the renormalization of $\overline{\Psi}{\small\Gamma\overleftrightarrow{\nabla}}\Psi$ at one-loop level.
• Figure 2: Diagrams for perturbative contributions. The last two diagrams are related to the counterterms.
• Figure 3: Diagrams for $m\langle\bar{q}q\rangle$ contributions.
• Figure 4: Diagrams for $\langle\bar{q}q\rangle\langle\bar{q}Gq\rangle$ and $m\langle GG\rangle\langle\bar{q}q\rangle$ contributions. The $\langle \mathcal{O}_8\rangle$ refers to condensates like $\langle\bar{q}\nabla\nabla\nabla\nabla\nabla q\rangle$, $\langle\bar{q}GG\nabla q\rangle$, $\langle\bar{q}DDDGq\rangle$, etc., which contribute to $\langle\bar{q}q\rangle\langle\bar{q}Gq\rangle$ and $m\langle GG\rangle\langle\bar{q}q\rangle$ via Eq. \ref{['d8_expansion']}.
• Figure 5: NLO GSR mass predictions for $2^{++}$ states extracted from $J^{\{\mu\alpha\}}$; the central values of the QCD parameters are used. Dashed lines in (a) and (b) mark the optimal value of $s_0$, as well as the corresponding $\chi^2$ and predicted mass. For $\bar{u}d$ configuration, the LO and NLO perturbative contributions after Gaussian transformation are shown in (d), where $s_0=3\,\text{GeV}^2$ is used; the NLO correction is considerable.